Answer:
To find the real and complex roots of the function f(x) = x^4 + 74x^2 - 567, we can use the quadratic formula to solve for x^2 and then find the square roots of the solutions.
Let's first set y = x^2, then we can rewrite the function as:
f(y) = y^2 + 74y - 567
Using the quadratic formula, we can find the roots of this quadratic equation as:
y = (-74 ± sqrt(74^2 + 4*567))/2y = (-74 ± sqrt(5929))/2y = (-74 ± 77)/2
So, we have two solutions for y:
y1 = -75.5
y2 = 1.5
Now, we can find the square roots of these solutions to get the values of x:
x1 = ±sqrt(-75.5) = ±8.68i
x2 = ±sqrt(1.5) = ±1.22
Therefore, the complex roots of the function are x1 = 8.68i and x2 = -8.68i, and the real roots are x3 = 1.22 and x4 = -1.22